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Journal of Physics A Mathematical and General

Geometric `tempus' for a class of ergodic classical systems

Christian G. Schroer

Published Jun 21, 1996• 1 citation
Publisher

Abstract

Using the symplectic formalism of classical mechanics, a smooth holonomy effect is identified in a special class of classical ergodic Hamiltonian systems (generalized canonical families) that are cycled adiabatically. It is a smooth shift in a time-resembling observable `tempus' that is canonically conjugated to the phase space volume contained in the energy shell. The curvature 2-form of the adiabatic connection obtained is closely related to Hannay's and the Robbins - Berry 2-form. The measurability of the `tempus' is discussed.

DOI
10.1088/0305-4470/29/12/031

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Using the symplectic formalism of classical mechanics, a smooth holonomy effect is identified in a special class of classical ergodic Hamiltonian systems (generalized canonical families) that are cycled adiabatically. It is a smooth shift in a time-resembling observable tempus' that is canonically conjugated to the phase space volume contained in the energy shell. The curvature 2-form of the adiabatic connection obtained is closely related to Hannay's and the Robbins - Berry 2-form. The measurability of the tempus' is discussed.
The ``geometrical phase factor,'' found by Berry in his study of the quantum adiabatic theorem, has a classical analogue discovered recently by Hannay and Berry himself. This classical analogue consists of an extra angle swept by the system in its classical evolution during an adiabatic excursion. As in Berry's phase, Hannay's angle reflects a nontrivial holonomy of the system and it has a purely geometric character. In this paper we calculate Hannay's angles for the classical Grassmannian system associated to a quantum spin in a magnetic field. We also show their close relation to the corresponding quantum-mechanical phase of Berry for the same system.
General time-dependent, bilinear classical Lagrangian mechanics can be interpreted as a charged particle interacting with a gauge scalar and vector potential. For an arbitrary discrete number of degrees of freedom, there are enough symplectic symmetries in classical mechanics to make this system canonically identical to the most general discrete spectrum quantum system. It follows that the general invariance group of the Schr\"odinger system of equations is the symplectic group Sp(2N), not U(N). Several examples show the physical naturalness of the classical coordinates. In the adiabatic limit Berry's phase is identified as the cumulative change of the angle of action-angle coordinates of a normal mode. A generalization of Berry's phase, the time component of the adiabatic connection and its associated electric curvature, is shown to be on an equal footing with the usual Berry phase.
This Theorem establishes that the Q5 transport holonomy and the Berry holonomy of the canonical Bloch system are the same closed-loop invariant, not merely comparable representatives. Under the conditions that the reduced transport family is cyclic and nondegenerate, the induced Bloch loop is well-defined and closed, and adiabatic transport is defined on the eigenbundle: Hol_{Q5} := ∮μ(Q)dλ = Hol_Berry = γ_Berry (mod 2π) The proof follows directly from T41: the rotational trace one-form μ(Q)dλ differs from the Berry connection only by an exact differential, which integrates to zero on a closed loop. The gauge-invariant holonomy lemma confirms that the loop integral is independent of admissible gauge transformations preserving the cyclic transport class. The distinction between the Q5 transport phase and geometric Berry phase is therefore one of representation, not of underlying invariant. Status: Gauge-invariant holonomy lemma exact, follows from exact-differential argument of T41. Holonomy equivalence exact at the closed-loop level, independent of gauge. Inherits the normalization condition required for the T41 connection-level identification. No additional assumptions introduced beyond T41. Dependencies: T39, T40, T41.
Berry connections are shown to be invariant under the action of symmetry groups on parameter space. This observation allows one to use the theory of invariant Yang-Mills potentials to evaluate these connections without computing the instantaneous energy eigenstates. Hamiltonians belonging to the algebra of a Lie group are examined in this light. They provide a wide class of systems admitting nontrivial adiabatic holonomy. The special case of the generalized harmonic oscillator is analyzed and the SO(2,1) invariance of the associated Berry connection is exhibited.
We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.
We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.